# Entropy Theory and its Application in Environmental and Water Engineering

- ID: 2329572
- February 2013
- 662 Pages
- John Wiley and Sons Ltd

Entropy Theory and its Application in Environmental and Water Engineering responds to the need for a book that deals with basic concepts of entropy theory from a hydrologic and water engineering perspective and then for a book that deals with applications of these concepts to a range of water engineering problems. The range of applications of entropy is constantly expanding and new areas finding a use for the theory are continually emerging. The applications of concepts and techniques vary across different subject areas and this book aims to relate them directly to practical problems of environmental and water engineering.

The book presents and explains the Principle of Maximum Entropy (POME) and the Principle of Minimum Cross Entropy (POMCE) and their applications to different types of probability distributions. Spatial and inverse spatial entropy are important for urban planning and are presented with clarity. Maximum entropy spectral analysis and minimum cross entropy spectral analysis are powerful techniques for addressing a variety of problems faced by environmental and water scientists and engineers and are described here with illustrative examples.

Giving a thorough
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Preface, xv

Acknowledgments, xix

1 Introduction, 1

1.1 Systems and their characteristics, 1

1.1.1 Classes of systems, 1

1.1.2 System states, 1

1.1.3 Change of state, 2

1.1.4 Thermodynamic entropy, 3

1.1.5 Evolutive connotation of entropy, 5

1.1.6 Statistical mechanical entropy, 5

1.2 Informational entropies, 7

1.2.1 Types of entropies, 8

1.2.2 Shannon entropy, 9

1.2.3 Information gain function, 12

1.2.4 Boltzmann, Gibbs and Shannon entropies, 14

1.2.5 Negentropy, 15

1.2.6 Exponential entropy, 16

1.2.7 Tsallis entropy, 18

1.2.8 Renyi entropy, 19

1.3 Entropy, information, and uncertainty, 21

1.3.1 Information, 22

1.3.2 Uncertainty and surprise, 24

1.4 Types of uncertainty, 25

1.5 Entropy and related concepts, 27

1.5.1 Information content of data, 27

1.5.2 Criteria for model selection, 28

1.5.3 Hypothesis testing, 29

1.5.4 Risk assessment, 29

Questions, 29

References, 31

Additional References, 32

2 Entropy Theory, 33

2.1 Formulation of entropy, 33

2.2 Shannon entropy, 39

2.3 Connotations of information and entropy, 42

2.3.1 Amount of information, 42

2.3.2 Measure of information, 43

2.3.3 Source of information, 43

2.3.4 Removal of uncertainty, 44

2.3.5 Equivocation, 45

2.3.6 Average amount of information, 45

2.3.7 Measurement system, 46

2.3.8 Information and organization, 46

2.4 Discrete entropy: univariate case and marginal entropy, 46

2.5 Discrete entropy: bivariate case, 52

2.5.1 Joint entropy, 53

2.5.2 Conditional entropy, 53

2.5.3 Transinformation, 57

2.6 Dimensionless entropies, 79

2.7 Bayes theorem, 80

2.8 Informational correlation coefficient, 88

2.9 Coefficient of nontransferred information, 90

2.10 Discrete entropy: multidimensional case, 92

2.11 Continuous entropy, 93

2.11.1 Univariate case, 94

2.11.2 Differential entropy of continuous variables, 97

2.11.3 Variable transformation and entropy, 99

2.11.4 Bivariate case, 100

2.11.5 Multivariate case, 105

2.12 Stochastic processes and entropy, 105

2.13 Effect of proportional class interval, 107

2.14 Effect of the form of probability distribution, 110

2.15 Data with zero values, 111

2.16 Effect of measurement units, 113

2.17 Effect of averaging data, 115

2.18 Effect of measurement error, 116

2.19 Entropy in frequency domain, 118

2.20 Principle of maximum entropy, 118

2.21 Concentration theorem, 119

2.22 Principle of minimum cross entropy, 122

2.23 Relation between entropy and error probability, 123

2.24 Various interpretations of entropy, 125

2.24.1 Measure of randomness or disorder, 125

2.24.2 Measure of unbiasedness or objectivity, 125

2.24.3 Measure of equality, 125

2.24.4 Measure of diversity, 126

2.24.5 Measure of lack of concentration, 126

2.24.6 Measure of flexibility, 126

2.24.7 Measure of complexity, 126

2.24.8 Measure of departure from uniform distribution, 127

2.24.9 Measure of interdependence, 127

2.24.10 Measure of dependence, 128

2.24.11 Measure of interactivity, 128

2.24.12 Measure of similarity, 129

2.24.13 Measure of redundancy, 129

2.24.14 Measure of organization, 130

2.25 Relation between entropy and variance, 133

2.26 Entropy power, 135

2.27 Relative frequency, 135

2.28 Application of entropy theory, 136

Questions, 136

References, 137

Additional Reading, 139

3 Principle of Maximum Entropy, 142

3.1 Formulation, 142

3.2 POME formalism for discrete variables, 145

3.3 POME formalism for continuous variables, 152

3.3.1 Entropy maximization using the method of Lagrange multipliers, 152

3.3.2 Direct method for entropy maximization, 157

3.4 POME formalism for two variables, 158

3.5 Effect of constraints on entropy, 165

3.6 Invariance of total entropy, 167

Questions, 168

References, 170

Additional Reading, 170

4 Derivation of Pome-Based Distributions, 172

4.1 Discrete variable and discrete distributions, 172

4.1.1 Constraint E[x] and the Maxwell-Boltzmann distribution, 172

4.1.2 Two constraints and Bose-Einstein distribution, 174

4.1.3 Two constraints and Fermi-Dirac distribution, 177

4.1.4 Intermediate statistics distribution, 178

4.1.5 Constraint: E[N]: Bernoulli distribution for a single trial, 179

4.1.6 Binomial distribution for repeated trials, 180

4.1.7 Geometric distribution: repeated trials, 181

4.1.8 Negative binomial distribution: repeated trials, 183

4.1.9 Constraint: E[N] = n: Poisson distribution, 183

4.2 Continuous variable and continuous distributions, 185

4.2.1 Finite interval [a, b], no constraint, and rectangular distribution, 185

4.2.2 Finite interval [a, b], one constraint and truncated exponential distribution, 186

4.2.3 Finite interval [0, 1], two constraints E[ln x] and E[ln(1 - x)] and beta distribution of first kind, 188

4.2.4 Semi-infinite interval (0,8), one constraint E[x] and exponential distribution, 191

4.2.5 Semi-infinite interval, two constraints E[x] and E[ln x] and gamma distribution, 192

4.2.6 Semi-infinite interval, two constraints E[ln x] and E[ln(1 + x)] and beta distribution of second kind, 194

4.2.7 Infinite interval, two constraints E[x] and E[x2] and normal distribution, 195

4.2.8 Semi-infinite interval, log-transformation Y = lnX, two constraints E[y] and E[y2] and log-normal distribution, 197

4.2.9 Infinite and semi-infinite intervals: constraints and distributions, 199

Questions, 203

References, 208

Additional Reading, 208

5 Multivariate Probability Distributions, 213

5.1 Multivariate normal distributions, 213

5.1.1 One time lag serial dependence, 213

5.1.2 Two-lag serial dependence, 221

5.1.3 Multi-lag serial dependence, 229

5.1.4 No serial dependence: bivariate case, 234

5.1.5 Cross-correlation and serial dependence: bivariate case, 238

5.1.6 Multivariate case: no serial dependence, 244

5.1.7 Multi-lag serial dependence, 245

5.2 Multivariate exponential distributions, 245

5.2.1 Bivariate exponential distribution, 245

5.2.2 Trivariate exponential distribution, 254

5.2.3 Extension to Weibull distribution, 257

5.3 Multivariate distributions using the entropy-copula method, 258

5.3.1 Families of copula, 259

5.3.2 Application, 260

5.4 Copula entropy, 265

Questions, 266

References, 267

Additional Reading, 268

6 Principle of Minimum Cross-Entropy, 270

6.1 Concept and formulation of POMCE, 270

6.2 Properties of POMCE, 271

6.3 POMCE formalism for discrete variables, 275

6.4 POMCE formulation for continuous variables, 279

6.5 Relation to POME, 280

6.6 Relation to mutual information, 281

6.7 Relation to variational distance, 281

6.8 Lin’s directed divergence measure, 282

6.9 Upper bounds for cross-entropy, 286

Questions, 287

References, 288

Additional Reading, 289

7 Derivation of POME-Based Distributions, 290

7.1 Discrete variable and mean E[x] as a constraint, 290

7.1.1 Uniform prior distribution, 291

7.1.2 Arithmetic prior distribution, 293

7.1.3 Geometric prior distribution, 294

7.1.4 Binomial prior distribution, 295

7.1.5 General prior distribution, 297

7.2 Discrete variable taking on an infinite set of values, 298

7.2.1 Improper prior probability distribution, 298

7.2.2 A priori Poisson probability distribution, 301

7.2.3 A priori negative binomial distribution, 304

7.3 Continuous variable: general formulation, 305

7.3.1 Uniform prior and mean constraint, 307

7.3.2 Exponential prior and mean and mean log constraints, 308

Questions, 308

References, 309

8 Parameter Estimation, 310

8.1 Ordinary entropy-based parameter estimation method, 310

8.1.1 Specification of constraints, 311

8.1.2 Derivation of entropy-based distribution, 311

8.1.3 Construction of zeroth Lagrange multiplier, 311

8.1.4 Determination of Lagrange multipliers, 312

8.1.5 Determination of distribution parameters, 313

8.2 Parameter-space expansion method, 325

8.3 Contrast with method of maximum likelihood estimation (MLE), 329

8.4 Parameter estimation by numerical methods, 331

Questions, 332

References, 333

Additional Reading, 334

9 Spatial Entropy, 335

9.1 Organization of spatial data, 336

9.1.1 Distribution, density, and aggregation, 337

9.2 Spatial entropy statistics, 339

9.2.1 Redundancy, 343

9.2.2 Information gain, 345

9.2.3 Disutility entropy, 352

9.3 One dimensional aggregation, 353

9.4 Another approach to spatial representation, 360

9.5 Two-dimensional aggregation, 363

9.5.1 Probability density function and its resolution, 372

9.5.2 Relation between spatial entropy and spatial disutility, 375

9.6 Entropy maximization for modeling spatial phenomena, 376

9.7 Cluster analysis by entropy maximization, 380

9.8 Spatial visualization and mapping, 384

9.9 Scale and entropy, 386

9.10 Spatial probability distributions, 388

9.11 Scaling: rank size rule and Zipf’s law, 391

9.11.1 Exponential law, 391

9.11.2 Log-normal law, 391

9.11.3 Power law, 392

9.11.4 Law of proportionate effect, 392

Questions, 393

References, 394

Further Reading, 395

10 Inverse Spatial Entropy, 398

10.1 Definition, 398

10.2 Principle of entropy decomposition, 402

10.3 Measures of information gain, 405

10.3.1 Bivariate measures, 405

10.3.2 Map representation, 410

10.3.3 Construction of spatial measures, 412

10.4 Aggregation properties, 417

10.5 Spatial interpretations, 420

10.6 Hierarchical decomposition, 426

10.7 Comparative measures of spatial decomposition, 428

Questions, 433

References, 435

11 Entropy Spectral Analyses, 436

11.1 Characteristics of time series, 436

11.1.1 Mean, 437

11.1.2 Variance, 438

11.1.3 Covariance, 440

11.1.4 Correlation, 441

11.1.5 Stationarity, 443

11.2 Spectral analysis, 446

11.2.1 Fourier representation, 448

11.2.2 Fourier transform, 453

11.2.3 Periodogram, 454

11.2.4 Power, 457

11.2.5 Power spectrum, 461

11.3 Spectral analysis using maximum entropy, 464

11.3.1 Burg method, 465

11.3.2 Kapur-Kesavan method, 473

11.3.3 Maximization of entropy, 473

11.3.4 Determination of Lagrange multipliers ?k, 476

11.3.5 Spectral density, 479

11.3.6 Extrapolation of autocovariance functions, 482

11.3.7 Entropy of power spectrum, 482

11.4 Spectral estimation using configurational entropy, 483

11.5 Spectral estimation by mutual information principle, 486

References, 490

Additional Reading, 490

12 Minimum Cross Entropy Spectral Analysis, 492

12.1 Cross-entropy, 492

12.2 Minimum cross-entropy spectral analysis (MCESA), 493

12.2.1 Power spectrum probability density function, 493

12.2.2 Minimum cross-entropy-based probability density functions given total expected spectral powers at each frequency, 498

12.2.3 Spectral probability density functions for white noise, 501

12.3 Minimum cross-entropy power spectrum given auto-correlation, 503

12.3.1 No prior power spectrum estimate is given, 504

12.3.2 A prior power spectrum estimate is given, 505

12.3.3 Given spectral powers: Tk = Gj, Gj = Pk, 506

12.4 Cross-entropy between input and output of linear filter, 509

12.4.1 Given input signal PDF, 509

12.4.2 Given prior power spectrum, 510

12.5 Comparison, 512

12.6 Towards efficient algorithms, 514

12.7 General method for minimum cross-entropy spectral estimation, 515

References, 515

Additional References, 516

13 Evaluation and Design of Sampling and Measurement Networks, 517

13.1 Design considerations, 517

13.2 Information-related approaches, 518

13.2.1 Information variance, 518

13.2.2 Transfer function variance, 520

13.2.3 Correlation, 521

13.3 Entropy measures, 521

13.3.1 Marginal entropy, joint entropy, conditional entropy and transinformation, 521

13.3.2 Informational correlation coefficient, 523

13.3.3 Isoinformation, 524

13.3.4 Information transfer function, 524

13.3.5 Information distance, 525

13.3.6 Information area, 525

13.3.7 Application to rainfall networks, 525

13.4 Directional information transfer index, 530

13.4.1 Kernel estimation, 531

13.4.2 Application to groundwater quality networks, 533

13.5 Total correlation, 537

13.6 Maximum information minimum redundancy (MIMR), 539

13.6.1 Optimization, 541

13.6.2 Selection procedure, 542

Questions, 553

References, 554

Additional Reading, 556

14 Selection of Variables and Models, 559

14.1 Methods for selection, 559

14.2 Kullback-Leibler (KL) distance, 560

14.3 Variable selection, 560

14.4 Transitivity, 561

14.5 Logit model, 561

14.6 Risk and vulnerability assessment, 574

14.6.1 Hazard assessment, 576

14.6.2 Vulnerability assessment, 577

14.6.3 Risk assessment and ranking, 578

Questions, 578

References, 579

Additional Reading, 580

15 Neural Networks, 581

15.1 Single neuron, 581

15.2 Neural network training, 585

15.3 Principle of maximum information preservation, 588

15.4 A single neuron corrupted by processing noise, 589

15.5 A single neuron corrupted by additive input noise, 592

15.6 Redundancy and diversity, 596

15.7 Decision trees and entropy nets, 598

Questions, 602

References, 603

16 System Complexity, 605

16.1 Ferdinand’s measure of complexity, 605

16.1.1 Specification of constraints, 606

16.1.2 Maximization of entropy, 606

16.1.3 Determination of Lagrange multipliers, 606

16.1.4 Partition function, 607

16.1.5 Analysis of complexity, 610

16.1.6 Maximum entropy, 614

16.1.7 Complexity as a function of N, 616

16.2 Kapur’s complexity analysis, 618

16.3 Cornacchio’s generalized complexity measures, 620

16.3.1 Special case: R = 1, 624

16.3.2 Analysis of complexity: non-unique K-transition points and conditional complexity, 624

16.4 Kapur’s simplification, 627

16.5 Kapur’s measure, 627

16.6 Hypothesis testing, 628

16.7 Other complexity measures, 628

Questions, 631

References, 631

Additional References, 632

Author Index, 633

Subject Index, 639

Vijay P. Singh, Texas A & M University, USA